What does it mean to have epochs=30 in Keras' fit method given certain data?
Batch size and epochs are independent parameters - they serve very different purposes. Your main question as I understand it (and for general, non-library specific consumption) is what is an epoch and how is the data used for each epoch?

Simply put, an epoch is a single iteration though the training data. Each and every sample from your training dataset will be used once per epoch, whether it is for training or validation. Therefore, the more epochs, the more the model is trained. The key is to identify the number of epochs that fits the model to the data without overfitting.

Your explaination of how batch size affects the training process is correct but not relevant to the question since it has no relation to the epoch training iterations. That is not to say that these values should be considered independently since they have similar effects on the model training process.